Critical Hausdorff measure and its gauge in dimensions at least three

Determine whether the critical Hausdorff measure of the extreme-point set of a random countable stable zonotope with uniformly distributed directions satisfies \(\mathcal H^{(d-1)\alpha}(F_\alpha)=0\) for every \(d\geq2\), and, for \(d\geq3\), identify the correct gauge function governing its generalized Hausdorff measure.

Background

The paper establishes that the critical Hausdorff measure $\mathcal H^{(d-1)\alpha}(F_\alpha)$ is almost surely finite. In dimension two, known results for the range of a stable subordinator imply that the corresponding critical Hausdorff measure is almost surely zero, with a larger logarithmically corrected gauge producing a positive and finite measure.

The authors conjecture that vanishing of the critical measure holds in every dimension. For dimensions at least three, both this conjecture and the appropriate gauge function remain unresolved because the one-dimensional subordinator argument does not extend to the $(d-1)$-dimensional parameter space.